Abstract
Let \(X = G/K\) be a symmetric space of noncompact type. A result of Gelander provides exponential upper bounds in terms of the volume for the torsion homology of the noncompact arithmetic locally symmetric spaces \(\Gamma \backslash X\). We show that under suitable assumptions on \(X\) this result can be extended to the case of nonuniform arithmetic lattices \(\Gamma \subset G\) that may contain torsion. Using recent work of Calegari and Venkatesh we deduce from this upper bounds (in terms of the discriminant) for \(K_2\) of the ring of integers of totally imaginary number fields \(F\). More generally, we obtain such bounds for rings of \(S\)-integers in \(F\).
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References
Belabas, K., Gangl, H.: Generators and relations for \(K_2 O_F\). J. K-Theory 31(3), 195–231 (2004)
Bergeron, N., Venkatesh, A.: The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu 12(2), 391–447 (2013)
Borel, A.: Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 8(1), 1–33 (1981)
Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representations of reductive groups. Math. Surveys Monogr., vol. 67, American Mathematical Society (2000)
Borel, A., Yang, J.: The rank conjecture for number fields. Math. Res. Lett. 1, 689–699 (1994)
Brown, K.S.: Cohomology of groups, Graduate texts in mathematics, vol. 87. Springer, Berlin (1982)
Calegari, F., Venkatesh, A.: A torsion Jacquet-Langlands correspondence, preprint arXiv:1212.3847v1
Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras, Pure and Applied Mathematics, vol. 11. Wiley, NY (1962)
Gelander, Tsachik: Homotopy type and volume of locally symmetric manifolds. Duke Math. J. 124(3), 459–515 (2004)
Houriet, J.: Arithmetic bounds—Lenstra’s constant and torsion of \(K\)-groups. Ph.D. thesis, EPF Lausanne (2010)
Samet, I.: Betti numbers of finite volume orbifolds. Geom. Topol. 17(2), 1113–1147 (2013)
Serre, J.-P.: Cohomologie des groupes discrets. Prospects in mathematics. Ann. Math. Stud., vol. 70. Princeton Universtiy Press, pp. 77–169 (1971)
Serre, J.-P.: Trees, Springer (1980)
Soulé, C.: Groupes de Chow et \(K\)-théorie de variétés sur un corps fini. Math. Ann. 268, 317–346 (1984)
Soulé, C.: Perfect forms and the Vandiver conjecture. J. Reine Angew. Math. 517, 209–221 (1999)
Soulé, C.: A bound for the torsion in the \(K\)-theory of algebraic integers. Doc. Math. Extra, vol. Kato, pp. 761–788 (2003)
Weibel, C.: Algebraic \(K\)-theory of rings of integers in local and global fields. In: Friedlander, E.M., Grayson, D.R. (eds.) Handbook of \(K\)-theory, vol. 1. Springer (2005)
Acknowledgments
I would like to thank Akshay Venkatesh for his help and encouragement, and Mike Lipnowski, Jean Raimbault and Misha Belolipetsky for helpful comments.
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Supported by Swiss National Science Foundation, Projects number PA00P2-139672 and PZ00P2-148100.
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Emery, V. Torsion homology of arithmetic lattices and \(K_2\) of imaginary fields. Math. Z. 277, 1155–1164 (2014). https://doi.org/10.1007/s00209-014-1298-2
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DOI: https://doi.org/10.1007/s00209-014-1298-2